arXiv subjects

Félix Parraud

Publications and source records attributed to Félix Parraud.

11 recordsLinked to original sources

arXiv2026

Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{ρ,ζ}\colon |ζ|<ρ\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{ρ,ζ}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $δ>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq δ\right)\leq \left(C t/δ\right)^{N^{2/3}} \] for a constant $C=C_ρ$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

math.PR↗

arXiv2026

Asymptotic expansion for transport maps between laws of multimatrix models

We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.

math.PR↗

arXiv2025

A Haagerup inequality through the use of orthogonal polynomials

In this paper we prove that the Haagerup inequality for non-homogeneous polynomials in free semicircular variables of degree $n$ is optimal with a constant of order $n^{3/2}$. We also show an operator valued Haagerup inequality which improves on existing results. Our main tool to do so are free Chebyshev polynomials also known as $0$-Hermite polynomials.

math.OA↗

arXiv2025

Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices

Let $U^N$ be a family of $N\times N$ independent Haar unitary random matrices and their adjoints, $Z^N$ a family of deterministic matrices, and $P$ a self-adjoint noncommutative polynomial, i.e. for any $N$, $P(U^N,Z^N)$ is self-adjoint, $f$ a smooth function. We prove that for any $k$, if $f$ is smooth enough, there exist deterministic constants $α_i^P(f,Z^N)$ such that $$ \mathbb{E}\left[\frac{1}{N}\text{Tr}\left( f(P(U^N,Z^N)) \right)\right]\ =\ \sum_{i=0}^k \frac{α_i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal{O}(N^{-2k-2}) .$$ Besides, the constants $α_i^P(f,Z^N)$ are built explicitly with the help of free probability. As a corollary, we prove that given $α<1/2$, for $N$ large enough, every eigenvalue of $P(U^N,Z^N)$ is $N^{-α}$-close to the spectrum of $P(u,Z^N)$ where $u$ is a $d$-tuple of free Haar unitaries. We also prove the convergence of the norm of any polynomial $P(U^N\otimes I_M, I_N\otimes Y^M)$ as long as the family $Y^M$ converges strongly and that $M\ll N \ln^{-3}(N)$.

math.PR↗

arXiv2025

The free energy of matrix models

In this paper we study multi-matrix models whose potentials are perturbations of the quadratic potential associated with independent GUE random matrices. More precisely, we compute the free energy and the expectation of the trace of polynomials evaluated in those matrices. We prove an asymptotic expansion in the inverse of the matrix dimension to any order. Out of this result we deduce new formulas for map enumerations and the microstates free entropy. Our approach is based on the interpolation method between random matrices and free operators developed in [9, 31].

math.PR↗

arXiv2024

The spectrum of a tensor of random and deterministic matrices

We consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices and we show that its $L^p$-norm can be upper bounded, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M=\exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In particular this provides another proof of the Peterson-Thom conjecture thanks to the result of Ben Hayes. We also obtain similar results for polynomials in random and deterministic matrices. The approach that we take in this paper is based on an asymptotic expansion obtained by the same author in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $\mathcal{C}^*$-algebras.

math.PR↗

arXiv2024

Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times.

math.PR↗

arXiv2022

Convergence for noncommutative rational functions evaluated in random matrices

One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges asymptotically and can be described with the help of free probability. This paper aims to show that this can be extended to noncommutative rational functions, answering an open question by Roland Speicher. This paper also provides a noncommutative probability approach to approximating the free field. At the algebraic level, its construction relies on the approximation by generic matrices. On the other hand, it admits many embeddings in the algebra of operators affiliated with a $II_1$ factor. A consequence of our result is that, as soon as the generators admit a random matrix model, the approximation of any self-adjoint noncommutative rational function by generic matrices can be upgraded at the level of convergence in distribution.

math.OA↗

arXiv2021

On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Let $U^N = (U_1^N,\dots, U^N_p)$ be a d-tuple of $N\times N$ independent Haar unitary matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. In 1998, Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$ \frac{1}{MN} \text{Tr}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) , $$ and its limit when $N$ goes to infinity. If $f$ is seven times differentiable, we show that it is bounded by $M^2 \left\Vert f\right\Vert_{\mathcal{C}^7} N^{-2}$. As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Actually we show that if $U^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})$ converges almost surely towards its free limit.

math.PR↗

arXiv2021

Concentration estimates for random subspaces of a tensor product, and application to Quantum Information Theory

Given a random subspace $H_n$ chosen uniformly in a tensor product of Hilbert spaces $V_n\otimes W$, we consider the collection $K_n$ of all singular values of all norm one elements of $H_n$ with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of $W$ fixed and the dimension of $H_n$ and $V_n$ tending to infinity at the same speed in a paper of Belinschi, Collins and Nechita. In this paper, we provide measure concentration estimates in this context. The probabilistic study of $K_n$ was motivated by important questions in Quantum Information Theory, and allowed to provide the smallest known dimension (184) for the dimension an an ancilla space allowing Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where violation of MOE occurs.

quant-ph↗

arXiv2019

On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N\times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$\frac{1}{MN}\text{Tr}\left( f(P(X^N\otimes I_M,Z^{NM})) \right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2\left\Vert f\right\Vert_{\mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjørnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $β>0$ such that for any $\varepsilon_1<3+β)^{-1}$ and $\varepsilon_2<1/4$, almost surely for $N$ large enough, $$-\frac{1}{N^{\varepsilon_1}}\ \leq \| P(X^N)\| - \left\Vert P(x)\right\Vert \leq\ \frac{1}{N^{\varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^N\otimes I_{M_N},I_N\otimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjørnsen, where $M_N$ had size $o(N^{1/4})$.

math.PR↗